Fast enumeration for Grassmannian space

The Grassmanian G_q(n; k) is the set of all k-dimensional subspaces of vector space Fⁿq. The coding of elements of Grassmanian was considered in many papers [1]-[7], and has the application in network coding [8]-[19]. The enumerative coding of the elements of Grassmanian G_q(n; k) is association every element of the Grassmanian with its number, i. e. the number from [0;...; |G_q(n; k)| - 1]. The algorithm of enumerative coding of the elements of the Grassmanian, which has complexity O(nk(n - k) log n log log n) is presented in the paper [20]. We present the advanced algorithm of the enumerative coding of the elements of the Grassmanian, which has the complexity that does not exceed O(n²log²nloglog n). The advanced algorithm is based on the method of fast enumeration of combinatorial objects from the paper of B. Ryabko [21].